Inductance and resistance
The time constant of an inductance
L and a resistance
R is equal to
L / R, and represents the time to change the current in the inductance from zero to
E / R at a constant rate of change of current
E / L (which produces an induced voltage
E across the inductance).
If a voltage
E is applied to a series circuit comprising an inductance
L and a resistance
R, then after time
t the current
i, the voltage
vR across the resistance, the voltage
vL across the inductance and the magnetic linkage
yL in the inductance are:
i = (E / R)(1 - e - tR / L)
vR = iR = E(1 - e - tR / L)
vL = E - vR = Ee - tR / L
yL = Li = (LE / R)(1 - e - tR / L)
If an inductance
L carrying a current
I is discharged through a resistance
R, then after time
t the current
i, the voltage
vR across the resistance, the voltage
vL across the inductance and the magnetic linkage
yL in the inductance are:
i = Ie - tR / L
vR = iR = IRe - tR / L
vL = vR = IRe - tR / L
yL = Li = LIe - tR / L
Rise Time and Fall Time
The rise time (or fall time) of a change is defined as the transition time between the 10% and 90% levels of the total change, so for an exponential rise (or fall) of time constant
T, the rise time (or fall time)
t10-90 is:
t10-90 = (ln0.9 - ln0.1)
T » 2.2
T
The half time of a change is defined as the transition time between the initial and 50% levels of the total change, so for an exponential change of time constant
T, the half time
t50 is :
t50 = (ln1.0 - ln0.5)
T » 0.69
T Note that for an exponential change of time constant
T:
- over time interval
T, a rise changes by a factor
1 - e -1 (
» 0.63) of the remaining change,
- over time interval
T, a fall changes by a factor
e -1 (
» 0.37) of the remaining change,
- after time interval 3
T, less than 5% of the total change remains,
- after time interval 5
T, less than 1% of the total change remains.
Courtesy of:
http://www.bowest.com.au/library/formulae.html#13